Arctan and Polar Coordinates

Date: 03/09/99 at 14:22:44
From: Jim
Subject: What is an "arctan"?

As the last step in solving an example, we change 7.36R + 26.31U to 
polar form.

H = sqrt[(97.36)^2 + (26.31)^2] = 27.32
tan (th) = 26.31/7.36 = 3.575
th = arctan 3.575 = 74.37 deg

answer:  27.32 /_74.37 deg

What is an arctan and how did it do that?

Date: 03/09/99 at 17:22:02
From: Doctor Rick
Subject: Re: What is an "arctan"?

Let me guess what the problem means: 7.36 units to the right and 26.31 
units up. You could draw this on a graph as (x, y) = (7.36, 26.31), 
that is, x = 7.36 and y = 26.31. Then you want to find the polar 
coordinates of this point: the distance of the point from the origin 
(0, 0), and the angle that the line from (0, 0) to (x, y) makes 
with the positive x axis.

What you call H is the distance from the origin, or radius, of the 
point. You did not ask about it, so I guess you realize that this 
equation is the Pythagorean Theorem:

  r = sqrt(x^2 + y^2)

Now for the arctan. Do you remember what a tangent is? Draw a line from 
the point (x, y) perpendicular to the axis. This line, the x axis, and 
the line from (0, 0) to (x, y) form a right triangle:

                 (x, y)
                /|
               / |
              /  |
             /   |
            /    | y
           /     |
          /      |
         / th    |
        /________|...........> (x axis)
   (0, 0)   x    (x, 0)

The tangent of angle theta is the ratio of the opposite side of the 
triangle to the adjacent side, or y/x. The arctangent, also called the 
inverse tangent, of y/x is the angle theta that has this tangent. That 
is what is going on in your example.

  theta = arctan(y/x)

This works in quadrant I (right and up) where the angle is positive, 
and in quadrant IV (right and down) where the angle is negative, 
because the arctan function returns a value between -pi/2 and pi/2 
radians, or between -90 and 90 degrees. But if both x and y are 
negative (left and down), y/x is the same as if they were both 
positive. Also, we must be careful about what happens if x = 0, since 
we cannot divide by 0. So the rule for converting to polar coordinates 
has to be more complex:

if x > 0 then
  theta = arctan(y/x)
if x < 0 then
  theta = arctan(y/x) - 180 degrees
if x = 0 then
  if y > 0 then
    theta = 90 degrees
  if y < 0 then
    theta = -90 degrees
  if y = 0 then
    theta is indeterminate

Many computer languages have a function called atan2(y, x) that does 
all this automatically. Many scientific calculators have a R -> P key 
(rectangular to polar) that does the whole thing automatically, finding 
the radius as well as the angle.

I hope this helps you understand the method you are learning. Keep 
asking good questions like this!

- Doctor Rick, The Math Forum
  http://mathforum.org/dr.math/